Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes. In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem's parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube. Given that several such FO-definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.

翻译：近期多项研究受机器学习特别是聚类应用的驱动，开始探讨数据补全问题的参数化复杂度。值得注意的是，这些问题可以等价地转化为部分定义超立方体幂图的诱导子图上的经典图问题。本文延续这一新兴研究方向，通过研究该图类上的独立集问题展开深入探讨——该问题在数据科学领域以"多样性"之名被广泛研究。我们构建了该问题参数化复杂度的完整图景，并证明了其关于解规模与超立方体幂次之和的固定参数可解性。鉴于已有研究表明多个此类一阶逻辑可定义问题在该图类上具有固定参数可解性，我们自然追问：固定参数可解性能否扩展至所有一阶逻辑可定义问题？我们通过证明超立方体诱导子图上的一阶模型检测与一般图上的检测难度相当，对此问题给出了否定回答。